Automorphism groups of compact Riemann surfaces of genera three and four (Q921169)

Let X be a compact Riemann surface of genus \(\geq 2\), and let AG be a group of automorphisms of X. Since AG acts on the vector space of holomorphic differentials on X, it can be represented as a subgroup R(X,AG) of GL(g,\({\mathbb{C}})\). It arises a problem: Determine all subgroups of GL(g,\({\mathbb{C}})\) which are conjugate to R(X,AG) for some X and some AG. For \(g=2\), this problem was already solved by the first author. In this paper, the authors solved it for \(g=3\) and 4.